This corollary establishes the asymptotic optimality when correct models are contained in the candidate model set, which complements the asymptotic optimality in Theorem REF . It shows that when correct models are available in the candidate set, the squared loss of the JCVMA estimator of \({\mu }\) , namely, \(L_n(\widehat{{\bf w}})\) , is asymptotically negligible compared to that of any averaging estimator that assigns zero weights to the correct models. In conjunction with Theorem REF , this corollary suggests that JCVMA also provides good prediction when correct models are available, since it asymptotically assigns all weights to the correct models and outperforms any other averaging predictions that fail to include these models. Note that the asymptotic optimality in Corollary REF  concerns the squared loss of the averaging estimator of \({\mu }\)  but does not directly suggest the estimation efficiency of \({\beta }\) . Thus, it differs from the semiparametric efficiency bounds studied in [1]}.
